Words that almost commute

Abstract: The \emph{Hamming distance} $\text{ham}(u,v)$ between two equal-length words $u$, $v$ is the number of positions where $u$ and $v$ differ. The words $u$ and $v$ are said to be \emph{conjugates} if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\text{ham}(xy,yx)$ can take on is $0$, when $x$ and $y$ commute. But, interestingly, the next smallest value $\text{ham}(xy,yx)$ can take on is $2$ and not $1$. In this paper, we consider conjugates $u=xy$ and $v=yx$ where $\text{ham}(xy,yx)=2$. More specifically, we provide an efficient formula to count the number $h(n)$ of length-$n$ words $u=xy$ over a $k$-letter alphabet that have a conjugate $v=yx$ such that $\text{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Finally, we show that there is no one easily-expressible good bound on the growth of $h(n)$.